Finite Gr\"obner basis algebra with unsolvable nilpotency problem and zero divisors problem

TL;DR

This paper constructs two finite Gr"obner basis-defined algebras demonstrating the algorithmic unsolvability of nilpotency and zero divisor problems, providing negative answers to longstanding questions in algebra.

Contribution

It introduces explicit algebra examples with finite Gr"obner bases where key decision problems are algorithmically unsolvable, addressing open questions in algebraic theory.

Findings

First algebra has an unsolvable nilpotency problem.

Second algebra has an unsolvable zero divisor problem.

Provides counterexamples to algorithmic solvability in algebra.

Abstract

This work presents a sample constructions of two algebras both with the ideal of relations defined by a finite Gr\"obner basis. For the first algebra the question whether a given element is nilpotent is algorithmically unsolvable, for the second one the question whether a given element is a zero divisor is algorithmically unsolvable. This gives a negative answer to questions raised by Latyshev.